How it might have happened

The reason the little ice age happened seems to be a large set of stages and catalysts that all worked together to form the conditions needed for an event like this.
For the sake of brevity I am only going to be showing the most realistic and probable causes.

Solar Acitiviy and Orbital Position

There is a bit of a weak link between solar activity and the little ice age. Many studies have looked over sunspot records, isotopes generated by cosmic rays, and the large sections of decreased solar activity that happened during the mentioned period (1300-1850).
The Earth's atomosphere blocks a large amount of radiation even after the magnetic field. Protons interact with gasses in the atmosphere which turn into isotopes(unstable atoms which are often radioactive) like carbon-14 or beryllium. These isotopes fall to Earth and get mixed in with water and trees which scientist use to track solar activity over thousands of years.
Scientist observed a decrease in solar activity like sunspots and increase in isotopes around and during the little ice age; many would conclude that this would cause a decrease in temperature, although there was a 50 year "lag" from when the solar activity was observed and the little ice age may have started. "A 50-year response lag is possible but is not consistent with subsequent variations in inferred solar activity and average northern hemisphere temperature." (Quote: Wikipedia; Owens, M. J.; et al. Maunder Minimum article)

Even the amount of sunspots isn't very conlusive on whether or not they affected global temps. There seems to have been a higher rate, at some points, of sunspots; especially compared to modern day. None of them seem to have greatly changed anything. Quoting the research of Judith Lean, "... it is possible that TSI was actually higher in the Maunder Minimum than present-day levels, but uncertainties are high with best estimates of the difference between the modern-day TSI and the Maunder-Minimum TSI ..." (Quote: Wikipedia; Judith Lean, Sun-climate Relationship) They don't seems to have that much of an effect on the temps more specifically on the little ice age.

Volcanic Activity

During the little ice age there were many volcanic eruptions that happened, a good few including sulfer. Volcanic eruptions spew smoke and ash into the atmosphere which often greatly cools the Earth and affected areas down, also blocking sunlight. The sulfer is another great coolant, when interacting with sunlight in the stratosphere it can turn into sulfuric acid and reflect even more sunlight on top the smoke and ash.
A large majority of them happened in the 1300s, the starting point of the little ice age. Another bout of eruptions happened around the 1700s, nearing the end of the event. The effects of some even reaching the Americas, "The 1815 eruption of Tambora, also in Indonesia, blanketed the atmosphere with ash, and the following year came to be known as the Year Without a Summer, when frost and snow were reported in June and July in both New England and Northern Europe."

All of these are a huge contributer to the cooling of the Earth and Europe, and can maybe be attributed to causing the little ice age.

Other Notable things

1.Ocean circulation: Some people say the slowing of the AMOC could have changed temperatures.
Although it can lead to cooler temps, the changing of AMOC can also heat things up. There was a period of heat, called the "Medieval Warm Period" that happened before the little ice age. Both of these things may have been affected by currents.

2. Decreased population: During the middle ages, where the ice age happened, there were lots of wars, lots of sickness, and invasions. One thing most people remember is the black death, a plague that killed 1/3 of Europe, majorly bringing down the population. Also, the interaction and killing of American Natives brought down the temperatures a lot. "The elimination of nearly 55 million, or 90 per cent, of Indigenous people in the Americas during European colonization led to global climate change ... Colonization of the Americas at the end of the 15th century killed so many people, it disturbed Earth's climate, according to a new study by University College London." (John Hadden, Ice Age caused by Native Holocaust)

All of the major killings and genocides that happened during the middle ages and exploration.